**The Musical System of Archytas**

Robert Erickson

The music theory upon which our conceptions of ancient Greek music are based
starts with Aristoxenus who flourished around 318 B.C. Very little has been preserved of the
earlier music theory relating to the music of the Classic period of Greece, the music praised,
criticized and endlessly discussed in the works of such writers as Plato, Aristophanes and
Aristotle. If more music had been preserved this would have been no great loss. In this
absence our knowledge is based mostly upon opinions and observations which are often vague
and conflicting about technical matters. The *Orestes* fragment and the mathematical
formulations by Philolaus, Plato and Archytas are bits of solid evidence floating in a sea of
opinion.

Archytas' ratios have been commented upon since antiquity^{ (1)} but their
interrelationships have never been analyzed. I propose to show that Archytas' ratios present
an interrelated intervallic system, and that the structure of the system offers valuable evidence
about pre-Aristoxenian Greek music, especially about *harmonia*^{ (2)} and the technique of
transposition, features which later were developed into complete systems of octave
species and *tonoi*.^{ (3)}

Archytas' ratios were preserved by Ptolemy^{
(4)}, a set for each *genus*, as follows:

Enharmonic: 28/27 36/36 5/4

Diatonic: 28/27 8/7 9//8

Chromatic: 28/27 243/224 32/27

Each set represents the intervals of a tetrachord. If two tetrachords are separated by a tone, the conventional disjunctive, an octave scale can be constructed:

From A to c- in any of the above scales is the interval 7/6, (9/8 x 29/27 equals
7/6), a septimal minor third.^{ (5)} Therefore from F- down to an imagined lower D would be a 7/6
also. From A up to c in the enharmonic is a 6/5 minor third (9/8 x 28/27 equals 6/5)).
Therefore from the imagined lower D up to F would be a 6/5 minor third too, and the fifths D
to A and A to e would be constituted: 6/5 x 5/4 equals 3/2.^{ (6)} Tannery first suggested^{
(7)} that
Archytas ratios should be approached from this point of view, and that the fifth D to A is divided
by its harmonic mean F, the fourth A to d by its harmonic mean c-. Adopting Tannery's
suggestion we are able to set out the following framework in musical notation:

These calculations involving the harmonic mean place enharmonic *lichanos* (F)
and *trite* (c-), and give an important clue to Archytas' other interval divisions too. Lichanos is
the most sensitive tone in Greek tunings, for its placement determines the character of the
largest interval in the tetrachord. The placement of *trite* is equally important to Archytas, for
it creates both the 7/6 third and the *pyknon* interval 28//27, which appears in all three *genera*.

Significantly the division of the fourth and fifth is by the harmonic mean.^{ (8)} From
Archytas' writings this might well be expected, for it was he who renamed the subcontrary mean
harmonic because of its use in music, and he is responsible for the proof that no
supraparticular ratio can be divided into equal rational parts.^{ (9)} If superparticular ratios cannot
thus be divided it follows that for Archytas the most important method of division is the
harmonic.

It appears that superparticular ratios were felt to produce "natural" musical
intervals throughout the whole development of Greek theory from Pythagoras to Ptolemy. This
conflict between Aristoxenus and the harmonists is really a quarrel between practicality and
analytical precision.^{ (10)}

When Pythagoras divided the octave he discovered the ratios for fourth and fifth;
and these, 3/2 and 4/3, are superparticular. Hippasus and Philolaus show interest in
superparticular ratios and the harmonic mean, and the attention paid to superparticular ratios
and the harmonic mean, and the attention paid to superparticular ratios in the mathematical
theory of irrationals has a background of practical music theory.^{ (11)} The interval system
presented by Archytas' ratios harmonizes with his proof about the division of superparticular
intervals, for all of his chief intervals are derived through the harmonic mean.

Therefore, we may be sure that the framework D to A to d was a fully conscious theoretical creation. It is the consequence of Archytas' desire to extend the scope of harmonic division.

The framework is blended with that of the usual two tetrachords separated by a disjunctive tone: E to A, tone, B to E. Together they make a structure either of two fifths, D to A to e, or tone/fourth,tone/fourth, D, E, to A, B to e. (See below.)

This framework, already containing the higher d, can be made into a double system by the addition of G: E to A, B to e is the same pattern as D to G, A to d, only a tone higher. These seven tones are the "standing" tones of Archytas' double system.

If
diatonic and enharmonic are superimposed a new interval, the small second 10/9,
is revealed:

This important interval appears among the ratios (see Appendix B) of both
Didymus and Ptolemy. Its appearance in the enharmonic/diatonic mixture above is the first hint
that Archytas' ratios are somehow inter-related.

By combining diatonic and chromatic, it is possible to follow the development of
Archytas' idea. Ptolemy^{ (12)} tells us that Archytas said that he placed chromatic *lichanos* (F#) in
the relationship 256/243 to diatonic *lichanos* (G). Here is what happens when, following that
hint, Archytas' diatonic and chromatic *genera* are combined. The first
scale below shows the combination within the usual framework; the second gives a
new scale running from F# to f#.

The scale which I have extracted in the Pythagorean scale, the most important
and long-lived scale in music;^{ (13)} no doubt Archytas built his set of ratios in such a way as to
make this tuning available, for his F# of the chromatic was by design calculated not from A but
from G of the diatonic *genus*. Ptolemy disapproved but missed the point.

Clearly there is more to these ratios than the numbers for the three *genera* alone.
Already we have found two varieties of diatonic; and there are more: in all, three diatonics,
three chromatics and three enharmonics, all verified by later usage. In addition there are a
number of tunings probably important to Archytas but not confirmed by others. The chart below
superimposes all three sets of ratios for an overview of the main tunings available in the
system. In order to accommodate the various scales, the range has been extended. Remarks
beside each scale note identical or similar tunings by other composers. These tunings are all
listed in Appendix B. For the tunings of Aristoxenus, who calculated in distances rather that
ratios I have used the cents equivalents worked out by Winnington-Ingram in his "Aristoxenus
and the Intervals of Greek Music". The *pyknon*, referred to below is the group of narrow
intervals such as E to F- to F or, if uncleft, E to F or F# to G.

The system makes several other tetrachords available which, however, do not
work out to be complete octave scales of two identical tetrachords. It is quite likely that they
were intended by Archytas, and that they were used, for from what we know of practice, it was
not unusual to mix tetrachords of various "shades" and *genera*. They all involve tetrachord *synnemenon*, using either B or B-. All are diatonic.

I have not included any ratios for the much discussed Spondeion scale;^{ (14)} but two
of Aristoxenus' chromatic "shades" are unaccounted for, and they may present two versions
or approximations of some of the intervals of the Spondeion:

Chromatic

Malakon66 66 366Chromatic

Meliolion75 75 348

Now there is a tetrachord among Archytas' ratios with cents measurements as follows: 49, 92,
357. The 357 cent interval falls between the 366 and 348 of Aristoxenus with less than 10
cents difference either way. Archytas' 49 cents comes nowhere near 66 cents or 75 cents, but
Archytas' 49 plus 92 equals the interval 141 cents, and this is within 9 cents of Aristoxenus' 75
plus 75 cents equal to 150 cents. Archytas' tetrachord splits the difference between
Aristoxenus' Chromatic *Malakon* and Chromatic *Hemiolion*, and there is a fair presumption that
all three tetrachords represent a single type. The tetrachord below, using B- instead of B
for tetrachord synnemenon, may represent the *Spondeion*, with its special intervals, *spondaismyus*, *sklysis* and *ekbole*, as mentioned by Aristides and Bacchuis. Winnington-Ingram points out that Bacchius illustrated these intervals using the same part of the scale we
are considering, at the point of conjunction or disjunction of the tetrachords.^{ (15)}

*Ekbole*: 5 dieses *Spondaismus*: 3 dieses *Eklysis*: 3 dieses In Aristoxenian terminology a *diesis*
equals a "quarter-tone".

Three-quarters of a tone equals about 148 cents.

All of the main intervals in the Archytas system are derived through the harmonic
mean. The large intervals are derived as follows^{ (16)}:

The small intervals are also a result of manipulation of the harmonic mean:

The interval 243/224 is not superparticlar, and probably Archytas regarded it as
a limma (literally "the rest") in the same way that 256/243 is a *limma*. Ptolemy criticized
Archytas severely for this *"ekmelic*"interval^{ (17)} on the grounds that Archytas is measuring from *genus* to *genus* to make it and that it is no ratio. The interesting point, however, is that
Archytas manages to divide that interval 243/224 into two parts, 49 cents and 92 cents. I
believe the 92 cents part is the most significant, because it is auditorialy identical to 90 cents,
the Pythagorean *limma* of 256/243. In effect Archytas makes it possible to begin tetrachords
from either F or F# with a well known and traditional interval. This fact alone points toward a
system. Coupled with the documentation by the tunings of other composers, especially
Aristoxenus, I think we can be certain that Archytas' ratios were constructed in such a way that
they made and were meant to make an interrelated tuning system.

Archytas' System and the *Harmoniai* from Plato's *Republic*

The famous discussion from Book III of the *Republic* includes mention of a
number of *harmoniai*: Dorian, Phyrgian, Lydian, Mixolydian and Iastian. Music in these *harmoniai* is supposed to have strong psycho-physical effects. These effects, their, nature and
power, have been the subject of speculation ever since antiquity. With the help of Archytas'
system it is possible to reconstruct the actual sounds of the fourth century *harmoniai*. Fitting
the *harmoniai* to the system will give us not only an excellent auditory impression of the sounds
Plato heard - that would be worth something in itself - but show some of the reasons why
Archytas might have invented his system.

Plato's *harmoniai* were preserved by Aristides Quintilianus ^{ (18)}, as recipes (Dorian:
tone, diesis, diesis, diton, tone diesis, diesis, ditone) and in letter notation. They are different
from the octave scales of later Greek theory. Dorian and Phrygian have a range of a ninth
(compare Archytas' framework), others less than an octave. Some omit one or more tones,
and Iastian and Mixolydian have diatonic elements, even though the prevailing *genus* is
enharmonic.

Several of the *harmoniai* are said to be "*malakon*", an epithet later applied to
"soft" tetrachords, such as those containing 8/7 and 7/6 tones rather than 9/8 tones.^{ (19)} One of
the Lydians is called Syntonolydian, usually taken to mean high in pitch, or intense, or both.
Another of the Lydians is called "*chalaros*" - "low and lax", or "slack". From these terms it
appears that *harmoniai* may begin at various degrees of a tuning system. It may be only a
coincidence that Plato's Dorian and Phyrgian together require a range of a ninth, and that the
tuning system of his friend and colleague, Archytas, has a ninth for its framework. Could both
have been following some sort of current practice? Such as the Iasti-Aioli
tuning described below?

At any rate it appears that Archytas has created a system in which a number of *harmoniai* can be played, at various degrees and in various shadings, and which permitted easy
modulation by pitch degree and/or *genus* and "shade." Since Archytas' system and Plato's *harmoniai* are contemporary it seems reasonable to take them together to construct a model
of fourth century *harmoniai* in a fourth century tuning system.^{ (20)}

First the Iastian (Ionian), by way of example. Aristides recipe is *dieses*, *diesis*, *diton*, a three semitone interval (*trihemitone*) and a tone. He points out that "this scale was a
tone less than the octave". His terms are Aristoxenian: a *diesis* is a quarter of a whole tone; a *ditone*, two whole tones; *trihemitone*, a tone and a half. Practically, the *ditone* is a "large third",
either the Pythagorean major third or the 5/4 third. The *trihemitone* is any sort of minor third,
narrow or wide. A diesis may vary from about 45 cents to something rather large. The dieses
from Archytas' system are 28/27 and 36/35. The *dieses* make up the *pyknon*.

The Iastian *harmonia* has a tetrachordal structure of two conjunct fourths. All of
the variants in the example below preserve that structure, though the interior intervallic
relationships may change with transposition. A certain "shade" of the Iastian is mentioned, *malakon*, and this variety of Iastian is especially identified with drinking parties and soft living.
Plato considered it unfit for the ears of his warrior-trainees.

I have included one each of diatonic and enharmonic forms of the Iastian for comparison. The
reader may wonder which of the four enharmonics is the "right" tuning. My feeling is that all
of those shown, and probably such others as could be gotten from the system would have been
used, either for convenience or by design. The intonations change but the fundamental
structure is identifiable. The Iastian enharmonic #4 is my candidate for the sort of *malakon*
Plato was against, with its up to date 5/4 third and thoroughly unpythagorean upper tetrachord;
its uncleft *pyknon* leans it toward the diatonic.

Other Iastians are available in the system; I hope I have shown enough to
indicate the scope of possible modulation. The significant point is that modulation by pitch
degree often is accompanied by a shading of *genus*. The two types of modulation go together.

All of Plato's harmoniai fit into Archytas' system on at least two scale degrees,
some of them with a number of changes of *genus* and "shade." There are many more
modulatory possibilities than I have shown below, with only two positions for each of the
remaining *harmoniai*. From the internal structure of any *harmonia* the reader can easily work
out which system degrees it can transpose to.

Aristides gives the Dorian in this form. Later it runs from E to e.

"High and "dirge-like". This lower tetrachord is chromatic. Compare below.

The fact that the *harmoniai* can begin on various scale degrees brings up the
question: are they primitive octave species? I think not, even though some significant
correspondences appear. The seven octave species (see note #3) compare well with the *harmoniai* except for the beginning note of the Dorian.

These pitch correspondences are striking, but should not be taken as proof that *harmoniai* were octave species. When theorists such as Aristoxenus set out to systematize
scale theory it is possible either that the old *harmoniai* had been torn to tatters by modulation
and chromaticism or that the intervallic relationships proper to any *harmonia* were a matter of
melodic tradition.

There is less direct relationship between the *harmoniai* and the later tonoi, but
the concept of tones carries a connotation of a (theoretically) transposed system, and we can
view Archytas' system as a congeries of systems beginning on B, C-, D, E, F-, F, F# etc., and
integrated into a single whole. More than anything else this whole resembles one of those
"close-packed" diagrams discussed by Aristoxenus. Unfortunately, everything we know about
these "close-packed" diagrams comes from him, but what he says is tantalizing:

And in dealing with the affinity between scales and regions of the voice we must not follow the Harmonists in their endeavor at compression, but aim rather at the intermodulation of scales, but considering in what keys the various scales must be set so as to admit of intermodulation. We have shown in a previous work that, though as a matter of fact some of the Harmonists have touched on this branch of our subject in a purely accidental way, in connection with their endeavour to exhibit a close-packed scheme of scales, yet there has been no general treatment of it by a single writer belonging to this school. This position of our subject may broadly be described as the part of the science of modulation concerned with melody.

^{ (21)}

Certain characteristics of Archytas' system harmonize with the theory of high and
low tuning elucidated by Otto Gombesi^{ (22)}: fairly large number of *harmoniai* can begin on either
E or F-. However, there are several other "good" degrees, D, E, F-, and F#. The structure of
the system is especially designed for Pythagorean shadings from F#.

All in all the evidence shows that the *harmoniai* were neither embryonic octave
species nor *tonoi*, even though certain characteristics of their behavior in what appears to be
a "close-packed" system may have been significant for later developments.

To get back to firmer ground, we can say that we have here a modulating system
adapted to the fourth century *harmoniai*. This is implicit in the framework itself (refer back to
page 89 - 90.) The fixed notes indicate a two *mese* system, but since beginnings may be made
from a number of pitch degrees it is really a multiple-*mese* system. It allows any *harmonia* to
be modulated by pitch degree, and, as shown above, the change of pitch degree may be
accompanied by a change of "shade" or *genus*, depending upon the *harmonia* and the degree
chosen.

Much more art was involved that simple transposition by a fourth up or down.
A rich modulatory technique must be assumed, and this is strongly supported by the
contemporary literary evidence, where there are copious references to modulation and mixing
of the *genera*. Plato's criticisms of the kitharists who mix all the *harmoniai* together and create
a chromatic confusion are all the more instructive for being negative.^{ (23)} The patterns of such
elaboration are built into Archytas' system, and it looks very much as though it were designed
to accommodate the practice of those very musicians whose performances were so distasteful
to Plato.

This impression is strengthened by Ptolemy`s discussion of the practical tunings
of the kitharists.^{ (24)}

Archytas' system resembles a number of the tunings used by those kitharists, and it corresponds most closely to Iasti-Aeoli, at the most favored system.

The range of a ninth invites the thought that Archytas had in mind just such a
tuning convention when he designed his ratios. By the time of Ptolemy the enharmonic had
fallen out of fashion, and that is why Iasti-Aioli is oriented toward chromatic/diatonic. From
Aristoxenus we know that the enharmonic was old-fashioned when he was writing his *Harmonics*, and the fourth century literary evidence shows chromatic and diatonic as the "new
sound" during Archytas' lifetime too.

The significant parallels between Iasti-Aioli and Archytas - range, identity of
tetrachords, modulating system are strong enough so that at the very least we may be sure that
Archytas' ratios are numerical shorthand for a modulating tuning system. In both systems
tetrachord relationships are ordered in such a way that a diatonic/chromatic tuning mix is
procured. Archytas' system provides additionally a complete tuning basis for selected pairs of
(sometimes different) tetrachords. Presumably players always tuned for *genus* color,
modulatory possibilities and convenience. In Archytas' time these patterns provided the
materials from which various *harmoniai *were constructed. *Harmoniai* could be played from
various pitch degrees and with certain modifications of "hardness" and "softness". The amount
and variety of modulation presumably depended upon the number of pitches available on an
instrument. Apparently much was possible on a nine-string lyre tuned in Iasti-Aioli. It is not
unthinkable that, given the notorious conservatism of instrumental practice and technique,
Ptolemy's Iasti-Aioli represents a tuning practice hundreds of years old, and that it existed in
this form during Archytas lifetime.

From what has been said it is clear that I do not regard Archytas' system as a paper abstraction. That means that it would have to be tuned by an ordinary musician rather than a scientist. I believe it was fairly easy to tune, on say a lyre or a harp, only slightly harder than Pythagorean.

Pythagorean is tuned by fifths and fourths, what the Greeks called "tuning by
consonance".^{ (25)} For example, if one wished to tune the diatonic, E,F, G,A,B,C,D,E, one
proceeded this way:

Varieties of Enharmonic:

This would produce the Pythagorean diatonic with limmas of 256/243 for half-steps, provided the player had tuned well. F# and G# could be added in the same manner:

(B) c#

F# (F#)

It is likely that the tuning error was plus or minus more than four cents, since even on a
monochord the error cannot be reduced further than that.^{ (26)}

Archytas' diatonic could be tuned as follows :

The player sets only one interval by ear, D to F-. Even so, he is able to use an *emmelic* interval - he never tunes a *pyknon* interval directly. From a practical point of view this
is a most important consideration, for no man is going to be able to tune say a 19/18 interval.
Humans are very good at octaves, fifths and fourth, able to tune a third in a triad to get rid of
beats, but otherwise quite insensitive to the ratios of small intervals having large fraction ratios.

To complete the system the player tunes to complete a Pythagorean diatonic scale: F#,G,A,B,c#,d,e,f#. Since G,A,B,d and e are already tuned he needs only to tune F# and c#.

The best thing about this analysis of Archytas' ratios is that it has given us a way to reconstruct with certainty the actual sound of fourth century Greek music. With a little practice one ought to be able to compose other sources of Greek music, and this would be a valuable way to explore the possibilities of the system and its musical characteristics. The system in turn, might well help solve some problems of detail in the transcription and reconstruction of the antique musical fragments.

In summary we find that an analysis of Archytas' ratios reveals:

1. A tuning system which structures modulation

(a) by pitch

(b) by

genusor "shade"2. Which reflects practical music making

(a) it is easy to tune

(b) it makes available all the important intervals of

Greek music

3. Which includes many of the tunings duplicated by later theorists.

(a) Aristoxenus: 4

(b) Erastosthanes: 2

(c) Didymus: 1

(d) Ptolemy: 3

(e) Aristides Quintillianus:

Spondeion.4. Which provides, with Plato's

harmoniai, a theoretical model of thestructure of fourth century Greek Music.

Practical, elegant in design, it is worthy of the ingenuity of the man who
generalled a city, built a wooden dove that would fly, wrote musical and political theory and
solved one of the great mathematical problems of antiquity, the duplication of the cube. This
new knowledge gleaned from his ratios can be used to illuminate several dark places in the
history of fourth century Greek music and music theory. Linked to other studies our whole view
of ancient Greek music might be considerably widened.

(San Francisco: February/March/April 1965)

Appendix A - Intervals with their Equivalents in Cents

2/1 1200 7/6 267 13/12 139

3/2 702 8/7 231 16/15 112

4/3 498 9/8 204 19/18 94

81/64 408 10/9 182 256/243 90

5/4 386 11/10 165 20/19 89

6/5 316 12/11 151 28/27 63

32/27 294 243/224 141 36/35 49

Appendix B - Tuning Ratios From Antiquity

Appendix C:

1. *Ptolemaios und Prophyrios uber die Musik*, Ingemar During, Goteborg, 1942, especially Chapters
13 and 14. Aristoxenus, *Harmonics*, ed. H.S. Macran, Oxford, 1902, 52 ff. *Aristoxenus and the
Intervals of Greek Music*, R.P. Winnington-Ingram, C.Q. 26, 1932, pp. 203, 205, 206. *The Musical
Scales of Plato's Republic*, J.F. Mountford, C.Q. 17, 1923. "The Harmonics of Ptolemy," J.F.
Mountford, *Transactions of the American Philological Association,* LVII, 1926, p.81. Harmonielehre
der Pythagorer, B.L. van der Waerden, Hermes 78, 1943, p.185.

2. A *harmonia* is fundamentally a tuning (Grove Dictionary) the word may include the meaning of
"scale". A definition which is true to its meaning through Plato, and the sense in which the term will
be used in this paper is: an ordered array of tones. See also *Musical Thought in Ancient Greece*,
Edward A. Lippman, New York, 1964, Ch.I.

3. To Aristoxenus and Ptolemy octave species were octave segments cut successively from a two octave diatonic system; but species could also be chromatic or enharmonic. See Grove Dictionary p.775. The seven species of diatonic octaves are:

B c d e f g a b Mixolydian

c d e f g a b c

^{1}Lydiand e f g a b c

^{1}d^{1}Phrygiane f g a b c

^{1}d^{1}e^{1}Dorianf g a b c

^{1}d^{1}e1^{ }Hypolydiang a b c

^{1}d^{1}e1 f^{1 }g^{1}Hypophrygiana b c

^{1}d^{1}e1 f^{1 }g^{1}a^{1}Hypodorian

*Tonoi* and *octave species* are related. A *tonos* has the name of that species (of the octave) whose
characteristic series of intervals it brings within a central range of pitch, for example the central
octave of the Dorian tonos. It is reasonable to assume that the original purpose of the *tonoi* was
to bring the species within this range, and that the species received their names, not as segments
of the Great Perfect System, but as different series of intervals within the same range." *Grove*,
p.776. *Tonoi* are sometimes called "transposition scales" in older literature, but Gombosi, "Key,
Mode, Species", JASM, 4, 1951, has argued that
*tonoi* are prior to species. The whole question
is discussed later in this paper.

4. Ptolemaios und Prophyrios uber die Musik, p.47. Ptolemy is our source for most of the ratios of antiquity. All of the ratios mentioned in this paper are given below in Appendix B.

5. This interval is our "blue third", head in Jazz and folk music. For a discussion of its use and
characteristics see *Report on Organization in Auditory Perception*, P.C. Boomsliter and Warren
Creel, Albany, NY., 1962, and recorded examples accompanying the booklet.

6. Intervals are added together by multiplying their ratios. For convenience to the reader all the intervals mentioned in this paper and their equivalents in cents are listed in appendix A.

7. *Memoires Scientifiques*, III, M.P. Tannery, p.247.

8. Archytas' definition of the harmonic mean, Ancilla, Freeman, Oxford 1962: "by whatever part of itself the first term exceeds the second, the middle term exceeds the third by the same part of the third. In this proportion the ratio of the larger numbers is larger, and of the lower numbers less." e.g., 6,4,3; 6-4=2, 4-3=1, and 2:6=1:3; 6/4 is greater than 4/3.

9. Preserved by Boethius, Diels-Kranz A19.

10. Aristoxenus was the exception. His criticism of all his predecessors is based upon the conception of a "good enough" interval, and he was aware that more than proper tuning was involved in music. His recipes show that he accepted the same musical intervals - give or take a few cents - which are stated as ratios by others. See appendix B.

11. *Companion to the Pre-Socratic* Philosophers, K. Freeman, Oxford, 1959: Hippasus, p.86,
Philolaus, p.224. For relations between theory of irrationals see "Theaetetus and the Theory of
Numbers," A. Wasserstein, C.Q. 52, 1958, 172 ff.

12. *Ptolemaios und Porphyrios über die Musik*, p.47.

13. Until now the earliest complete statement of its ratios was thought to be by Plato in his *Timaeus*,
and a few scholars have even argued that the scale was invented by Plato himself.

14. *Musical Scales of Plato's Republic*, J.F. Mountford. "The Spondaeion Scale," R.P. Winnington-Ingram, C.Q. 22, 1928.

15. "Aristoxenus and the Intervals of Greek Music," p.205.

16. Significantly there are ten intervals, using numbers up to ten, an elaboration of the original *Tetraktys* of the *Decad*.

17. Ptolemaios und Porhpyrios uber die Musik p.48.

18. Aristides Quintillianus, De Musica ed. R.P. Winnington-Ingram, Teubner texts, 1963. "Aristides
Quintillianus," *Von der Musik*, Rudolph Schafke, Berlin, 1937, pp.192-3.

19. Plato's *Republic* p.399. *The Rise of Music in the Ancient World*. C. Sachs, New York: 1943,
p.212, 228. *Studies in Musical Terminology in Fifth Century Literature*, Ingemar During, Eranos
43, 1945, p.180.

20. J.F. Mountford first applied Archytas' basic set of ratios to Plato's *harmoniai*. See his Musical
scales of Plato's *harmoniai*. See his "Musical Scales of Plato's *Republic*".

21. *Harmonics*, p.170.

22. "New Light on Ancient Greek Music," O.Gombosi, Papers of the International Congress of
Musicology, New York, 1939. *Tonarten und Stimmungen der Antiken Musik*, Copenhagen, 1939,
Otto Gomvosi.

23. *Laws, *669; *Republic*, 397. See also During in *Studies in Musical Terminology*, for his translation
of the Pherecrates fragment quoted in Pseudo-Plutarch. The attitudes expressed are those of the
musically conservative party to which Plato belonged, and During dates the fragments to 410 B.C.,
at which time Plato was about 18 years old. He grew up hearing - and resisting - the new
chromatic/diatonic practice.

24. Ptolemaios und Porphyrios uber die Musik, pp. 200-215, especially p.208.

25.
The method is demonstrated in Aristoxenus' instructions for tuning dissonances. *Harmonics, 55.*

26. The best error on a monochord with a one meter string is about 4 cents, according to C.D.
Adkins, *Theory and Practice of the Monochord*, University Microfilm 64-3344, Ann Arbor. For
tuning ability by professional factory "fine tuners" who can tune within 2 cents only with a visual
device, and then only in the middle octave of the piano, see JASA Vol.33 #5, p.582. In the
experience of musicians, 2 cents is a distinguishable lower limit in the case of fourths, fifths and
octaves. Other intervals, major and minor thirds, sixths, remain acceptable, if the musical situation
permits, within a large cents range. Whole and half-tones may vary widely, again depending upon
the musical situation. Many of the intervals discussed in this paper may be heard in our present
day concert and popular music; the interval 7/6 is the "blue third"; 10/9 intervals are quite common,
8/7 somewhat rarer, but the Pythagorean third is as "natural" as our equal temperament third; as
to small intervals, our non-keyboard players use all sizes, the 256/243 and 16/15 approximations
being the most usual. Anyone may verify this for himself: tune some of the intervals you wish to
study, or have them tuned by professional tuner. Then listen carefully to the actual intervals sung
or played by musicians. Check back to the pre-tuned sounds.